Every p-subgroup is well-placed in some Sylow subgroup

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Statement

Statement that involves choosing a Sylow subgroup

Suppose G is a finite group, p is a prime number, and W is a Conjugacy functor (?) on the p-subgroups of G. Suppose H is a p-subgroup of G. Then, there exists a p-Sylow subgroup P of G such that H is a Well-placed subgroup (?) in P relative to the conjugacy functor W.

Statement that involves conjugating the subgroup

Suppose G is a finite group, p is a prime number, and W is a Conjugacy functor (?) on the p-subgroups of G. Suppose H is a p-subgroup of G contained in a p-Sylow subgroup Q. Then, there exists a subgroup K of G, that is well-placed in Q, and is conjugate to H in G.

Related facts

Facts used

  1. Conjugacy functor sends every subgroup to a normalizer-relatively normal subgroup: For any p-subgroup H, H \le N_G(W(H)).
  2. Sylow subgroups exist
  3. Sylow implies order-dominating

Proof

Given: A finite group G, a prime p, a conjugacy functor W on the p-subgroups. A p-subgroup H of G.

To prove: There exists a p-Sylow subgroup P of G such that H is well-placed in G relative to W.

Proof: Define the following ascending chain of subgroups H_i. H_0 = H, and H_{i+1} is a p-Sylow subgroup of N_G(W(H_i)). By fact (1), this chain of subgroups is an ascending chain of p-subgroups. Let P be a p-Sylow subgroup of G containing the union of the H_is (such a P exists by facts (2) and (3)). We now prove that H is well-placed in P. This follows easily from the definition.